Proof of Rolle's Theorem in One Dimension

By Extreme Value Theorem on a Compact Interval, $f$ attains a maximum and minimum on $[a,b]$. If both extrema occur at endpoints and $f(a)=f(b)$, then $f$ is constant and any $c\in(a,b)$ satisfies $f'(c)=0$. Otherwise, at least one extremum occurs at some $c\in(a,b)$, and then Fermat Stationary Point Criterion gives $f'(c)=0$.